Application of derivatives Questions and Answers

Suppose y sin x. What is the x-value between 2π and 47 corresponding to a y-value of -1?

| Convert the given polar coordinates into rectangular coordinates. (-2, π) = ([?], []) Enter

Let y = 5x². Find the change in y, Δy when x = 2 and Δx = 0.2 Find the differential dy when x = 2 and dx=0.2

Let y = tan(5x + 8). Find the differential dy when x = 4 and dx = 0.2 Find the differential dy when x = 4 and dx = 0.4

Find the slope of the tangent line to the curve 3x² – xy – 2y³ = 160 at the point ( – 4, — 4). -

The function f(x) = 6x +9-¹ has one local minimum and one local maximum. This function has a local maximum at x =_____ with value______ and a local minimum at x=____ with value________

1 as follows: Let f(x) 1.001 C Use linear approximation, i.e. the tangent line, to approximate and find the equation of the tangent line to f(x) at a "nice" point near 1.001. Then use this to 1 approximate 1.001 1

Use the chain rule to find the derivative of f(x) = 3e f'(x) = 4-975

3 3. Sketch the graph of a function that satisfy the following conditions: f(0) = 0, f continuous and even. f'(x)=2x if 0<x< 1, f'(x) =-1 if 1<x<3 and f'(x) =1 if x > 3

The circumference of a sphere was measured to be 84 cm with a possible error of 0.5 cm. Use linear approximation to estimate the maximum error in the calculated surface area. Estimate the relative error in the calculated surface area.

Find the linear approximation of f(x) = ln x at x = 1 and use it to estimate In(1.28). L(x) = In(1.28) ~

Use Newton's method to approximate a root of the equation 2x + 3x + 3 = 0 as follows. Let x₁ = 1 be the initial approximation. The second approximation x₂ is and the third approximation x3 is

A rotating light is located 12 feet from a wall. The light completes one rotation every 3 seconds. Find the rate at which the light projected onto the wall is moving along the wall when the light's angle is 25 degrees from perpendicular to the wall. light wall K

A street light is at the top of a 16 ft tall pole. A woman 6 ft tall walks away from the pole with a speed of 6 ft/sec along a straight path. How fast is the tip of her shadow moving when she is 45 ft from the base of the pole? ft sec

A room contains 8 people, each of a different age. The people leave the room in random order, one at a time. What is the probability that they leave the room in ascending order of their ages (youngest to oldest)? The probability that they leave the room in ascending order of their ages is (Round to six decimal places as needed.) SECCI

5. Suppose f is continuous on [0, 4], f(0) = 1, and 2 Sf'(x) ≤ 5 for all x in (0, 4). Show that 9≤ f(4) ≤21. (Justify your work).

A 9-digit phone number cannot start with 0 or 1. Assume that there are no restrictions on the remaining 8 numbers. How many telephone numbers are possible in which all 9 digits are different? The possible number of 9-digit telephone numbers where all the digits are different is.

Given that f (x) = 2(3-x)²(x+4) (x-1)³ Solve the following inequality f(x) ≥ 0 Include the following information: The boundary points and the multiplicity of each The intervals . The sign(positive or negative) of the rational expression in each interval The solution set in interval notation . The sketch of the graph of f(x) ●

f(x)=-3x²+2x g(x) = 2x+3 Find f-g and fg. Then, give their domains using interval notation. (f - g)(x) = [] Domain of f - g : (f*g)(x) = Domain of fig

Find an equation of the tangent line to the graph of y = g(x) at x = 2 if g(2) = -4 and g'(2) = 3. (Enter your answer as an equation in terms of y and x.)

Find the magnitude and the direction of the resultant vector. (-6, -2) w A. 8.06, 60.26° B. 19.42, 34.51° C. 8.06, 29.74° D. 19.42, 55.50⁰ (10, 9)

Gravel is being dumped from a conveyor belt at a rate of 50 cubic feet per minute. It forms a pile in the shape of a right circular cone whose base diameter and height are always equal. How fast is the height of the pile increasing when the pile is 11 feet high? Recall that the volume of a right circular cone with height h and radius of the base r is given by 1 V = πr²h

dy Use implicit differentiation to find given the equation sin(xy) = y5. dx

The cost of a trip is $3.15 plus $0.95 for every kilometre that you ride on a bus. Which of the following expressions gives the cost of a trip of & kilometres? OC(k)= ($3.15+ $0.95)k OC(k)=$3.15+ $0.95 OC(k)= $3.15+ $0.95k OC(k) $3.15k + $0.95

If the tangent line to y = f(x) at (4, 2) passes through the point (0, 1), find f(4) and f'(4). (4) = f'(4) =

Let F(x) = f(x³) and G(x) = (f(x))³ . You also know that a¹ = 6, f(a) = 3, f'(a) = 6, f'(a) = 3 and G'(a) = Then F'(a) =

(10 points) A ladder 10 m long is leaning against a wall. If the base of the ladder is sliding away from the wall at the rate of 1 m per second, at what instantaneous rate, d will the top of the ladder be moving dt' when the base of the ladder is 6 m from the wall?

(10 points) A spherical balloon is inflated so that its volume is increasing at the rate of 20 cubic cm per second. What is the instantaneous change in surface area, ds of the balloon increasing when the radius is dt 40 cm? [Use V 4 == 3 r³ and S= 4².]

An oil spill is pouring oil onto a sea in the shape of a circle. The slick is growing at a rate of 150 m2/s. How fast is the radius changing when the radius is 25 m? T m/s O ² m/s 3 m/s 3π m/s

For the real-valued functions g(x)=x²-5. and h(x)=√x-2, find the composition g h and specify its domain using interval notation.

For the real-valued functions g(x) = 4x-1 and h(x)=√x+5, find the composition g h and specify its domain using interval notation.

A company manufactures 2 models of MP3 players. Let x represent the number (in millions) of the first model made, and let y represent the number (in millions) of the second model made. The company's revenue can be modeled by the equation R(z,y) = 140z + 170y - 2z² - 4y² - ry Find the marginal revenue equations R₂(z,y) = Ry(x, y) = 0 and We can achieve maximum revenue when both partial derivatives are equal to zero. Set Rz Ry = 0 and solve as a system of equations to the find the production levels that will maximize revenue. Revenue will be maximized when: X= y= =

The point A(3, 9) lies on the curve y=x² 40 If B is the point (x, x^2), find the slope of the secant line AB for the following values of x. Use all decimal places in your response. If x= 3.1, the slope of AB is: A/ If x= 3.01, the slope of AB is: A If x= 2.9, the slope of AB is: A/ If x= 2.99, the slope of AB is: A/ Based on the above results, guess the slope of the tangent line to the curve at A(3, 9):

Write an inequality to describe the region. The region between the yz-plane and the vertical plane x = 3. 00<x<3 0 < x² + y² + z² < 3 0<z <3 0<y<3 none of these

If Á = 2 +33 + 12kland B-3+43-6 are the two vectors, find the resultant vector. O 5+7+6 O 5² 73 +6 O 573 6 →> Osi+73-6

For the real-valued functions g(x)=x² +1. and h(x)=x-5, find the composition g h and specify Its domain using interval notation.

f(x) = -3x+4 g(x)=√4x-3 Find f.g and f+g. Then, give their domains using interval notation.

Find an equation of the sphere that passes through the point (1, 6, 3) and has center (3, 5, -5).

Find the scalar and vector projections of b onto a. a = (6, 7, -6) b = (3,-1, 1) scalar projection of b onto a vector projection of b onto a

In a composite function, If f[g(x)] = x=g[f(x)] then Of(x) and g(x) must be same functions Of(x) and g(x) is not a function Of(x) and g(x) are inverses Of(x) and g(x) must be different functions

Choose the best description of the normal line. a line parallel to a given line or curve a line with a zero slope C C a line with an undefined slope C a line perpendicular to a given line or curve

Find the extreme values of the function on the given interval: 16 f(x) = x² + ¹6; 1 ≤ x ≤4 1 and 4 1 and 2 2 and 4 1, 2, and 4

Find a + b, 9a + 7b, lal, and la - bl. (Simplify your vectors completely.) a = 9i - 8j + 7k, b = 7i - 9k a + b 11 9a + 7b = |a| = la - bl =

The conjugate of a complex number a + bi is 25+ 3i. What is a + bi? - 25 + 3i O25-3i O-25-3i 25+ 3i

Which of the following statements is the most correct statement regarding the critical points of the following function. 3 y = ax³ + bx² + cx+d there can be any number of critical points there are no critical points there will be a minimum of 2 critical points there will be a maximum of 2 critical points

Describe the surface in R3 represented by the equation x + y = 4, This is the set {(x, 4-x, z)|x ER, ze Ry which is a horizontal plane that intersects the xy-plane in the line y = 4 - x, z = 0. This is the set {(x, 4-x, z)|x ER, ZER) which is a vertical plane that intersects the xy-plane in the line y = 4 - x, z = 0. This is the set {(x, 4-x, z)|x ER, ZER3 which is a vertical plane that intersects the xz-plane in the line y = 4 - x, z = 0. This is the set {(x, 4-x, z)|x ER, ZERY which is a horizontal plane that intersects the xz-plane in the line y = 4 - x, z = 0. This is the set {(x, y, 4 - x - y) lx ER, YER) which is a vertical plane that intersects the xy-plane in the line y = 4 - x, z = 0.

If k is positive in y = f(x) + k, then shift the graph of the parent function by k units. upward downward right left

Find the slope of the tangent to the curve y = 1+ln z at the point (1, 1). 0 F N O 1 O o 1

Let f(x) 10x+12 point (0,5) is given by y m = b - 7e". Then the equation of the tangent line to the graph of f(x) at the = mx + b for

- Determine the slope of the tangent to the graph of y = (x² + 3x − 2)(7 - 3x) at (1,8).